Chapter 1 (Preliminaries)

Applied Calculus &

Analytical Geometry

About the course!

Books to follow!

• Course Book– CALCULUS by Thomas 11th Edition

(Compulsory)

• Reference Book– Calculus by Howard Anton 7th edition

(Optional)

Grading Policy

• Assignment 20%

• Quizzes 10%

• Midterm Exam 20%

• Final Exam 50%

Requirements

• Course Text book (Thomas Calculus 11th ed)• Separate Register only for Calculus I

(Because only those interested student’s questions will be entertained who will work continuously!!!)

What is Calculus?

• Calculus is the mathematics of Change.• Applied when we have

– Motion– Growth– Variable forces producing acceleration

• Invented to meet mathematical needs of scientists of 16th and 17th centuries.– The science of that time was mechanical in

nature

What is Calculus?

• The key elements were put in place, independently, by Newton (1642-1727) and Leibniz (1646-1716).

• The range of applications of calculus to ‘real world problems’ is vast and growing all the time.

• Two types– Differential calculus– Integral calculus

Differential Calculus

• Differential calculus deals with problems of “rates of change”– Finding slopes of curves– Velocities and accelerations of moving bodies– Find firing angles to have greatest range– Predict time when planets would be closest

together or farther apart

Integral Calculus

• Integral calculus deals with “determining a function from information about its rate of change”– Future location of a moving body– Areas of irregular regions in the plane– Measure lengths of curves– Find volumes and masses of arbitrary solids

Learning Calculus

• Read the text (Most Important)• Do the assignments as follows

– Sketch diagrams wherever possible, it will help in enhancing your visualization ability

– Write your solutions in connected step by step logical fashion, as if you are explaining to someone else

– Think about why each exercise is there? Why was it assigned? How is it related to the other assigned exercises?

Last words before start

• Please be in time (Don’t come late!)

• Keep your mobiles off (or silent) during class

• Don’t miss any class

• Solve assignments yourself !

• Handover assignments on time to get full grade!

• Don’t miss any quiz

Let us Start !!!

Chapter 1

Preliminaries

Topics to be covered

• Real Numbers and Real Line

• Lines, Circles and Parabolas

• Functions and their Graphs

• Identifying Functions• Combining functions, shifting and scaling graphs

• Trigonometric Functions

1.1-Real Numbers and the Real Line

The Real Line

Rules for Inequality

Main Subsets of Real Line

Rational and Irrational Numbers

Irrational Numbers

Interval

Example (Solving Inequality)

Absolute Value Function

Properties of Abs( )

Example

Exercise 1.1Practice Exercise (5—34 all odd problems)

Announcement

Please bring Low Priced Text book

• Thomas calculus (11th edition) and

• Separate Register for Calculus with you

On next

With your name written on it!

Every such student will be given 10 marks as part of (quiz)!!!!

Cartesian Coordinates in the Plane

Signs in Quadrants

Increments

Slope

Equation of Straight Line (Point Slope form)

Example

A Line through two points

Slope Intercept form

Example

Parallel and Perpendicular Lines

• Two lines are parallel if they have same slope. Conversely, if two lines have same slope then they are parallel.

• Two lines are perpendicular if product of their slopes is -1.That is,

m1 * m2 = -1

Distance formula in plane

Example

Equation of Circle

Example

Finding a circle’s radius and center

Interior and Exterior

Parabolas

Parabolas arise as graphs of:

ax2+bx+c = 0

Graph of eq y=ax2

Graph of ax2+bx+c

Role of a

Example

Exercise 1.2

Assignment #

Dead Line()

Ex 1.1 ( 7,9,11,15,17,25,27,31,33)

Ex 1.2 ( 1 to 45 and 53 to 60 odd)

Write down your name, subject, Section, ID, Assignment #

Function and their Graphs

Mapping

Some examples

f(x)= x+5

f(x)=x2+5x

f(x)= Sin(x)

f(x)=cos(x)

f(x)=5+sin(x)

Domain and Range

y = f(x)

DomainSet from where function (f) accepts values

Range (Co domain or Image set)

Set on which function (f) maps those values

Always remember that we shall consider Real values only not complex values throughout this course (Cal 1 and Cal 2)

Example

Example

1.

Greatest Integer Function

Least Integer Function

Writing formula for piecewise defined functions

Exercise 1.3

Identifying Function and Mathematical Models

Types of Functions

• Linear functions• Power functions• Polynomials• Algebraic functions• Trigonometric functions• Exponential functions• Logarithmic function• Transcendental functions

Linear functions

Power functions

Case 1 (xa ,a=n, a positive integer)

Case 2 (xa ,a=-1 or a=-2)

Case 3 (xa ,a=1/2,1/3,2/3,3/2)

Case 4 (Polynomial)

Case 5 (Rational Functions)

Summary

Summary

Graphs of Algebraic Functions

Graph of Sine and Cosine

Graph of Exponential Functions

Graph of Logarithmic Functions

Increasing and Decreasing functions

Even and Odd functions

Examples of Even and Odd functions

Even

Odd

Proportionality

Example

Mathematical Model

Example of proportionality

Exercise 1.4(1--30)

Combining Functions; shifting and Scaling

Example

Composite Functions

Example

Shifting Graph

Example

Scaling and Reflecting Graph

Example

Example

Ellipses

Exercise 1.5(1--50)

Trigonometric Functions

Conversion from Radian to Degree and from Degree to Radian

Example

Angle Convention

Important Quotients

Important Table to Cram !!!

Periodic Functions

Another to Cram (Also for Limits) !

Periods of Trigonometric Functions

Even and Odd

Identities

Addition Formulas

Double Angle Formulas (Imp)

Half Angle Formula (Imp)

The Law of Cosine

End of Chapter # 1